Let's work through the algebraic expression step-by-step:
We are given the expression [tex]\(3a \cdot a^2 \times 4a^5 \times a \cdot x^4 \times a^2\)[/tex].
First, let's address the constants and coefficients:
- The constants are [tex]\(3\)[/tex] and [tex]\(4\)[/tex].
- We multiply these constants together: [tex]\(3 \times 4 = 12\)[/tex].
Next, let's consider the terms involving [tex]\(a\)[/tex]:
- We have [tex]\(a\)[/tex] with exponents: [tex]\(a^1\)[/tex], [tex]\(a^2\)[/tex], [tex]\(a^5\)[/tex], [tex]\(a^1\)[/tex], and [tex]\(a^2\)[/tex].
- To combine these, we add the exponents together:
[tex]\[
1 + 2 + 5 + 1 + 2 = 11
\][/tex]
Now, let's consider the terms involving [tex]\(x\)[/tex]:
- We have [tex]\(x\)[/tex] with exponent [tex]\(x^4\)[/tex].
- Since there is only one term involving [tex]\(x\)[/tex], the exponent remains [tex]\(4\)[/tex].
Putting it all together:
- The combined coefficient is [tex]\(12\)[/tex].
- The combined power of [tex]\(a\)[/tex] is [tex]\(11\)[/tex].
- The combined power of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
Therefore, the simplified expression is:
[tex]\[
12a^{11}x^{4}
\][/tex]
This gives us the final answer as [tex]\(12a^{11}x^4\)[/tex].
